Realistic Math Education

I had the pleasure of attending the 5th International Realistic Mathematics Education (RME) 2015 Conference in Boulder, Colorado at the University of Colorado. Prior to attending the conference I did not know too much about RME. I knew that people that I admired in the field of mathematics education (Lucy West, Toni Cameron, Cathy Fosnot) were greatly influenced by the principles of RME and the Freudenthal Institute (FI).  Below is brief history, as posted on the Freudenthal Institute - USA:

In 2003 the Freudenthal Institute - USA (FI-US) was established as a result of the collaboration that had existed for years already between the Wisconsin Center for Education Research at the University of Wisconsin-Madison in the USA and the Freudenthal Institute. In Fall 2005, FI-US relocated to the University of Colorado at Boulder, USA.

The Freudenthal Institute for Science and Mathematics Education (FI) aims to improve education in the fields of arithmetic, mathematics, and the sciences, with a focus on primary, secondary and vocational education. FIsme works towards achieving these objectives through research, teaching, curriculum development and services.

What does any of this have to do with developing test-savvy math students, you may inquire?  Well the institute is grounded in the belief that learners need to consistently and deliberately experience the application of mathematics in order to learn how to mathematize their world...real and imagined.  RME is also grounded in the idea that learning needs to include informal, pre-formal and formal applications.  To emphasis this point the researchers at FI use an iceberg as a metaphor.  The top of the iceberg is what some people would call, "the math" or the abstractions or mathematic notation. But researchers at FI insist that we need to pay special attention to what in underneath the top of the iceberg.  Beneath the iceberg is where conceptual  understanding is developed and it is the majority of the iceberg of understanding.

So where does this leave testing and preparing for high-stakes standardized testing?  In my opinion...the same place it was before.  The concepts of RME are not new and have been influencing classroom instruction for decades.  The challenge was and remains to be...knowing that learners need significant time with the informal and pre-formal ideas and structures of mathematics.  Time and experiences that help to build towards the formalization of mathematical idea, without letting go of what is beneath the iceberg.  

Is it a matter of trust? Trusting that if we spend more time beneath the iceberg, with appropriate tools, models and tasks, that we will not drown.  Is it a matter of mindset?  Believing that some people can get the math and others cannot, so it does no make sense to focus so much time beneath the iceberg?  

I am not sure.  I just know that what we have been doing has not produced the results that we are looking for in terms of students being able to see mathematics in their world and use mathematics to describe their world.  

What can we do?

1. Critically and strategically examine our curriculum programs to note how much of our time is spent beneath the "iceberg" on given topics.
2. In teacher teams, vertically and horizontally across grade-levels, develop and examine our "icebergs" to note areas that need additional support and understanding from a teacher content stand-point.
3. Expand upon on our notions of assessment to include more experiences that demand the application of mathematical ideas in a realistic and sometimes messy ways.

Your thoughts on this topic would be greatly appreciated.

The 40 Year Effect!

Over the last year I have have the pleasure of working with Dr. Gregory of Dr. Gregory & Associates at an elementary school where we both do some consultancy work. He is constantly talking about the "40 Year Effect" that our instruction has on our students.  Dr. Gregory advocates that teachers should plan instruction experiences that stay with learners for 40 year, not 40 days or 40 months. Listen to the podcast below to find out more about the "40 Year Effect" and how you can achieve it.

I mention the 40 year effect, because I have been working with some adults who are looking to get into the renewable energies field, but did not meet the requirements in a certificate program. In order to gain the skills needed to meet the requirements they had to take part in a 5-week Math Boot Camp and I was their instructor. Initially, I was told that the concepts that I would be required to teach would be high school level mathematics, but in reality the mathematics concepts I actually had to teach would normally be introduced and taught between grades 3 to 7.  Topics such as area, perimeter and volume, operations with fractions, using measurement tools, decimals and percents, and ratios and proportions. Much of my experience mirrored what I see at the K - 12 level.  

As a teacher:

• I did not feel that I had to time to adequately prepare. The opportunity to work with these students came along rather suddenly and I had 2 days to plan out the 5-weeks. 
• I did not have access to any assessment data until after I met my students.  When I the opportunity came up for me to see the assessment data we began to realize that the assessment and the online tool for assembling the assessment data were inadequate.
• I was given a "curriculum" and asked to teach it. The "curriculum" consisted of 100 PowerPoint slides broken down by topics to teach each week. Upon review of the first 10 slides, I found 3 mathematical error or imprecise use of mathematical language. 
• The "curriculum" did not math the assessment tools. I realize rather quickly that in order for students to "progress" in the Math Boot Camp they had to make gains in the online assessment tool. However, the online assessment tool did not match the flow of the "curriculum".  For example, If a student tested in at Level 5, then they would begin the Math Boot Camp focused on ratios and proportions, but the "curriculum" did not address that topic until week 4.
• I was not told that my students would have a final exam until 4 before the last class. The director decided that the students needed to take a final exam that was separate from the online assessments that they had been taking. I saw the final exam 2 days before the last class.

My students:

• Reflected a very wide range of mathematical understandings. This group had people from all walks of life...a nurse, a lawyer, a carpenter, a recent high school graduate, a computer programmer, a handyman, a solider in the U.S. Army, etc. Most of my students were math phobic. Most of my students did not like to communicate their understanding...they just wanted to get to the answer. Most of my students were very procedural in their reasoning. Most of my students could "see" and make sense of numbers when the ideas were related to money, but had great difficulty when the numbers were in a different context. 
• Thought that I taught differently than anything they had ever experienced because I asked them to question, use more than one way to find a solution, work in groups, use mathematical sketches to visualize problems and used concrete materials to "show" and act out mathematical concepts. Most of my students lacked a deep conceptual understanding of the mathematics that they were being asked to use.

The "40 Year Effect" via student comments:

"Christine, I never knew what Pi meant. I never understood the relationship between the diameter and the circumference. I never understood that the circumference was the perimeter of a circle. Thanks to the (hands-on) activity that we did...now I will never forget."

"Christine, I can see the benefit of understanding now and not just knowing a procedure. I was trying to explain how to do something to another students, but realized that I did not understand enough or another way to explain things and they could not get it my way. I only knew how to get the answer my way, but I did not really understand what I was doing."

"Thanks for teaching us so many ways to solve one problem. I understand now and I have options. I now solve one way and check with another way and I know that I am correct. My teachers in school only taught me one way and if I did not get it, then I was stuck."

"When my son was in the 3rd grade he tested off the charts and was placed into a talented and gifted program. But where I come from, 'talented and gifted' programs are just codes for 'CEO training' programs. My son is brilliant and a little quirky, but in this 'talented and gifted' program they killed his creativity. He had to find solutions to way that they showed him or he was wrong. He now hates math is is very phobic about it. He doubts his abilities and his intelligence."

"Christine you taught me how to measure. I never knew. I thought that I could not ever get it. I never knew that it was so simple. It seemed so complicated before. I can measure now and I can teach my son."

"Christine, I had such a fear of math. I chose to become a social working, because I could avoid math. Now here I am trying to do better in life, change careers and it is the math that is hold me back, but at least now I know that I can do it." 

"I did not like to explaining part of the class. I can get the answer, but Christine insisted that we explain things." 

"You told us to sketch it out. I would not have gotten this one wrong if I drew it out. OMG, I can believe I left out the calculations for the area of the second window. I just did 1 window and the area of the wall." 

"I knew that I was going to fail the final, but I was courageous and came anyway. I tried. I just need more time to get this stuff. Thanks for helping me get this far. The online tutoring sites (NutShell Math via Cosmeo, Khan Academy) that we were using...are they going to be available even after the class is over?"

 What is your math "40 Year Effect" going to be?

Today was the "Big Day"

"Our students tested today.  They were as most kids are on standardized days...they were focused.  Of course they said the test was easy."  - CMK, May, 2011

I started writing this post almost a year ago, but was never inspired to finish it.  I am not really inspired right now either.  What I am is another adjective...frustrated.  Most of my schools are focused on...guess what...testing.  They say, "We have six-weeks, how do I best prep my students?"

Everyone is looking for a magic pill to give students, so that they know and understand all of the content that we are spoon feeding them.  It won't work.  I have hope that with the coming of the Common Core State Standards educators are prompted to take another look at how we prepare students for standardized testing. 

More test-prep is not the answer.  Test-savviness is an alternative.  We have to build thinking and reasoning into the test-prep process and it needs to be grounded in research-based instructional practices that raise the cognitive demand levels of students.  I recently did a workshop on the test-savvy strategies and the following is what a participant said and I thought with a sigh of relief, "Ahhh, she got it."

"I have to admit that I was expecting something totally different when I heard the words "test savvy." My first impression was that we'd focus on test sophistication skills and strategies such as bubbling, process of elimination, navigating the test, etc. So you can imagine how impressed and eager I was when I learned that you would equip us with tools and strategies to develop kids' conceptual understanding and deepen their overall thinking.

All of the principles and strategies we learned are so valuable in delivering meaningful and engaging instruction. I really enjoyed the hands-on nature of the PD. There are many parallels between this professional development session and my SIOP training. This is profound because when used together, these strategies will enhance my effectiveness in teaching ELLs. The PD offered a great deal of strategies so I will choose a few of the strategies I think my students would benefit from most to implement throughout the rest of the year. We've already launched "Word Problem Rework" in the class and it has been so eye opening to see what my students are capable of and where their misconceptions lie. Thanks so much Christine!" - NYC Teacher

A final thought from Dan Meyer @ TED:

Reading Levels of Word Problems

I have been exploring word problems and how to use reading comprehension strategies in math class.  I have been looking at ways to address comprehension and the importance to explicit instruction in the understanding, usage and application of mathematical vocabulary.  I will discuss those areas in upcoming posts.  In this post, I would like to discuss reading levels of word problems.

Look at the problem below:

"On Monday your snail made a slime trail 2 inches long. On Tuesday the slime trail was 4 inches long. If this same progress continues, how long will the slime trail be on Friday?"

This task came from the K - 2 section from Exemplars.  When I informally surveyed a few teachers  most felt that this problem would fit into the 3 - 5 grade level span.  When you analyze the task for grade level readability you get the following results:

Generated using: http://www.online-utility.org/english/readability_test_and_improve.jsp

Looking at the "Gunning Fog" index we can say that the average child, who started school in Kindergarten, would have to be in 7th grade to easily understand the text on the first reading. The grade level ranges, according to the second set indexes, are from the middle of the 3rd grade to the first few months of the 9th grade.

So what about the students who are not on grade level or ELL students?  These students are what Phil Daro, one of the write of the Common Core State Standards for Mathematics, would call the, "canaries in the mineshaft."   They provide a spotlight on issues the "average" student is able to coverup, because he/she knows how to "do school" and get the right answers.  

Word problems can be overwhelming and require us to take the time to provide instructional support at the reading comprehension level to enable students to fully understand problem and mathematics in context.

One strategy that can be used is to understand the math story by Retelling, before looking at the question that is being asked.  Tell students to explain the story without the use of numbers.  For example, the problem cited above might be retold in the following manner:

"Some snails were sliding along leaving a trail of slime. Yuck!  They went a little way on Monday, then went a little farther on Tuesday."

Another strategy that can be used is Visualization.  In reading visualization involves taking a text line by line (sentence by sentence) or paragraph by paragraph and seeing and describing the picture that can be painted from the words.  This strategy is a precursor to making mathematical sketches.  When we visualize we see, hear, smell and feel the story as it unfolds.  We talk about what has to be present in the environment, and consequently we eliminate erroneous ideas about the context of the problem.  This is an activity that is best when done collaboratively.  

Citing the word problem above, we might see a snail.  This might lead into a conversation about what a snail looks likes and what slime looks like.  And for many students, this is a necessary step if they have never had the opportunity to observe a snail in action.  The visualization may lead to deciding if this snail was a pet in a container or free in the wild.  In this case, students might conclude that it is a pet snail, due to the fact that we had to see the same snail on Monday and Tuesday.

A word of caution...reciting or listing the events in a problem does not constitute comprehension.  A child can read a book and tell you the sequence of events, but it does not mean that they comprehend the story.  Understanding what is read is a complex task, but a doable task when students are provided the time, structure, opportunities to collaborate, appropriate scaffolds and tools to be successful.  

Below are some additional resources to help you think more this issue and how to help students understand word problems:

1. "Literacy in the Secondary Classroom" Presented by Lucy West, Susan Radley Brown, Howard Segan and Anne Burgunder
2. Phil Daro's video interview on "Against Answer-getting"
3. "Using Reading Strategies in Math" from Reading Recovery
4. From Reading to Math: How Best Practices in Literacy Can Make You a Better Math Teacher, Grades K-5
5. Comprehending Math: Adapting Reading Strategies to Teach Mathematics, K-6